The Classification of the Finite Simple Groups, Number 3
The Classification of the Finite Simple Groups, Number 3

Author: Daniel Gorenstein

Publisher: American Mathematical Soc.

Published: 1994

Total Pages: 446

ISBN-13: 9780821803912

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Examines the internal structure of the finite simple groups of Lie type, the finite alternating groups, and 26 sporadic finite simple groups, as well as their analogues. Emphasis is on the structure of local subgroups and their relationships with one another, rather than development of an abstract theory of simple groups. A foundation is laid for the development of specific properties of K-groups to be used in the inductive proof of the classification theorem. Highlights include statements and proofs of the Breol-Tits and Curtis-Tits theorems, and material on centralizers of semisimple involutions in groups of Lie type. For graduate students and research mathematicians. Annotation copyrighted by Book News, Inc., Portland, OR

The Classification of the Finite Simple Groups, Number 5
The Classification of the Finite Simple Groups, Number 5

Author: Daniel Gorenstein

Publisher: American Mathematical Soc.

Published: 1994

Total Pages: 482

ISBN-13: 0821827766

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The fifth volume of the study proves two, and part of the third, of the planned five stages for the generic cast of the classification of finite simple groups. The main result is that either G has a p-uniqueness subgroup for some prime p, or that G has a neighborhood of semisimple subgroups that demonstrate certain properties in common with those in target simple groups G*. All this is preparation for the final stages, which are expected to deduce that G is about the same as G* for some known simple G*. Stay tuned. Perhaps an index will be deemed meet when the final answers are revealed. Annotation copyrighted by Book News, Inc., Portland, OR

The Classification of the Finite Simple Groups, Number 2
The Classification of the Finite Simple Groups, Number 2

Author: Daniel Gorenstein

Publisher: American Mathematical Soc.

Published: 1994

Total Pages: 246

ISBN-13: 9780821803905

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The second volume of a series devoted to reorganizing and simplifying proof of the classification of the finite simple groups. In a single chapter, it lays the groundwork for the forthcoming analysis of finite simple groups, beginning with the theory of components, layers, and the generalized Fitting subgroup, which has been developed largely since Gorenstein's basic 1968 text and is now central to understanding the structure of finite groups. Suitable as an auxiliary text for a graduate course in group theory. Member prices are $35 for individual and $47 for institutions. Annotation copyright by Book News, Inc., Portland, OR

The Classification of Finite Simple Groups
The Classification of Finite Simple Groups

Author: Michael Aschbacher

Publisher: American Mathematical Soc.

Published: 2011

Total Pages: 362

ISBN-13: 0821853368

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Provides an outline and modern overview of the classification of the finite simple groups. It primarily covers the 'even case', where the main groups arising are Lie-type (matrix) groups over a field of characteristic 2. The book thus completes a project begun by Daniel Gorenstein's 1983 book, which outlined the classification of groups of 'noncharacteristic 2 type'.

The Finite Simple Groups
The Finite Simple Groups

Author: Robert Wilson

Publisher: Springer Science & Business Media

Published: 2009-12-14

Total Pages: 310

ISBN-13: 1848009879

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Thisbookisintendedasanintroductiontoallthe?nitesimplegroups.During themonumentalstruggletoclassifythe?nitesimplegroups(andindeedsince), a huge amount of information about these groups has been accumulated. Conveyingthisinformationtothenextgenerationofstudentsandresearchers, not to mention those who might wish to apply this knowledge, has become a major challenge. With the publication of the two volumes by Aschbacher and Smith [12, 13] in 2004 we can reasonably regard the proof of the Classi?cation Theorem for Finite Simple Groups (usually abbreviated CFSG) as complete. Thus it is timely to attempt an overview of all the (non-abelian) ?nite simple groups in one volume. For expository purposes it is convenient to divide them into four basic types, namely the alternating, classical, exceptional and sporadic groups. The study of alternating groups soon develops into the theory of per- tation groups, which is well served by the classic text of Wielandt [170]and more modern treatments such as the comprehensive introduction by Dixon and Mortimer [53] and more specialised texts such as that of Cameron [19].

Finite Simple Groups
Finite Simple Groups

Author: Daniel Gorenstein

Publisher: Springer Science & Business Media

Published: 2013-11-27

Total Pages: 339

ISBN-13: 1468484974

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In February 1981, the classification of the finite simple groups (Dl)* was completed,t. * representing one of the most remarkable achievements in the history or mathematics. Involving the combined efforts of several hundred mathematicians from around the world over a period of 30 years, the full proof covered something between 5,000 and 10,000 journal pages, spread over 300 to 500 individual papers. The single result that, more than any other, opened up the field and foreshadowed the vastness of the full classification proof was the celebrated theorem of Walter Feit and John Thompson in 1962, which stated that every finite group of odd order (D2) is solvable (D3)-a statement expressi ble in a single line, yet its proof required a full 255-page issue of the Pacific 10urnal of Mathematics [93]. Soon thereafter, in 1965, came the first new sporadic simple group in over 100 years, the Zvonimir Janko group 1 , to further stimulate the 1 'To make the book as self-contained as possible. we are including definitions of various terms as they occur in the text. However. in order not to disrupt the continuity of the discussion. we have placed them at the end of the Introduction. We denote these definitions by (DI). (D2), (D3). etc.

The Classification of the Finite Simple Groups, Number 9
The Classification of the Finite Simple Groups, Number 9

Author: Inna Capdeboscq

Publisher: American Mathematical Society

Published: 2021-02-22

Total Pages: 520

ISBN-13: 1470464373

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This book is the ninth volume in a series whose goal is to furnish a careful and largely self-contained proof of the classification theorem for the finite simple groups. Having completed the classification of the simple groups of odd type as well as the classification of the simple groups of generic even type (modulo uniqueness theorems to appear later), the current volume begins the classification of the finite simple groups of special even type. The principal result of this volume is a classification of the groups of bicharacteristic type, i.e., of both even type and of $p$-type for a suitable odd prime $p$. It is here that the largest sporadic groups emerge, namely the Monster, the Baby Monster, the largest Conway group, and the three Fischer groups, along with six finite groups of Lie type over small fields, several of which play a major role as subgroups or sections of these sporadic groups.

Classes of Finite Groups
Classes of Finite Groups

Author: Adolfo Ballester-Bolinches

Publisher: Springer Science & Business Media

Published: 2006-07-10

Total Pages: 391

ISBN-13: 1402047193

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This book covers the latest achievements of the Theory of Classes of Finite Groups. It introduces some unpublished and fundamental advances in this Theory and provides a new insight into some classic facts in this area. By gathering the research of many authors scattered in hundreds of papers the book contributes to the understanding of the structure of finite groups by adapting and extending the successful techniques of the Theory of Finite Soluble Groups.

The Subgroup Structure of the Finite Classical Groups
The Subgroup Structure of the Finite Classical Groups

Author: Peter B. Kleidman

Publisher: Cambridge University Press

Published: 1990-04-26

Total Pages: 317

ISBN-13: 052135949X

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With the classification of the finite simple groups complete, much work has gone into the study of maximal subgroups of almost simple groups. In this volume the authors investigate the maximal subgroups of the finite classical groups and present research into these groups as well as proving many new results. In particular, the authors develop a unified treatment of the theory of the 'geometric subgroups' of the classical groups, introduced by Aschbacher, and they answer the questions of maximality and conjugacy and obtain the precise shapes of these groups. Both authors are experts in the field and the book will be of considerable value not only to group theorists, but also to combinatorialists and geometers interested in these techniques and results. Graduate students will find it a very readable introduction to the topic and it will bring them to the very forefront of research in group theory.

A Course on Finite Groups
A Course on Finite Groups

Author: H.E. Rose

Publisher: Springer Science & Business Media

Published: 2009-12-16

Total Pages: 314

ISBN-13: 1848828896

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Introduces the richness of group theory to advanced undergraduate and graduate students, concentrating on the finite aspects. Provides a wealth of exercises and problems to support self-study. Additional online resources on more challenging and more specialised topics can be used as extension material for courses, or for further independent study.